- A vector is a quantity that has both magnitude and direction, represented by an arrow.
- The magnitude (length) of a vector AB is written as ||AB|| and can be calculated as √((x2-x1)2 + (y2-y1)2) for vectors in the Cartesian plane.
- Common vector operations include addition, subtraction, scalar multiplication, and determining the resultant. Chasles' Rule states that AB + BC = AC.
- Vectors can be equal, opposite, collinear, orthogonal, or the zero vector (with magnitude 0 and no direction).
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. Scalar Vector
• A quantity is scalar if
• it has magnitude only.
• A quantity is vector if
• it has magnitude and direction.
• 2 parallel lines have
• the same direction.
3. Vectors
• Two vectors are equal if and only if the
quantities they represent have:
• 1) the same magnitude
• 2) the same direction
• A vector with magnitude 0 and no direction
is also possible.
• It is called the zero vector, 0.
• The opposite of AB is –AB or BA
4. Magnitude
• The size of AB, called its magnitude, or
norm, is written ║AB║
• ║AB║= √((x2-x1)2 + (y2-y1)2)
• If v (c,d), then║v║= √((c)2 + (d)2)
• A unit vector has length 1.
• A zero vector has length 0.
5. Exam QuestionGiven the three vectors u, v, and w.
v = (-2, -3)
u and w are represented in the Cartesian plane below:
x
y
u
w
Which of the following statements is TRUE?
A)
v
and -
u
are opposite.
B)
u
and
v
are equivalent.
C)
w
and (
v
+
w
) are perpendicular.
D)
u
and 3
v
are collinear.
6.
7. Relations Betwixt 2 Vectors
• If 2 vectors are perpendicular, they are
called orthogonal.
• Two vectors that are parallel are also
collinear.
• If one vector can be expressed
• in terms of another,
• (4,6) = 2(2,3), e = 2f,
• they are linearly dependent.
8. Exam Question
Quadrilateral RSTU is a parallelogram and M is
the point of intersection of its diagonals.
Antoine lists the following vector operation
statements:
S
R
T
U
M
1)
MU2SRST
2)
SM2URUT
3)
RTRURS
4)
0MUMSMRMT
5)
RTSTSR
Which of these statements are true?
A) 1, 2 and 3 only C) 2, 4 and 5 only
B) 1, 2 and 5 only D) 1, 3 and 4 only
9. Exam Question
Givena andb , two vectors illustrated below.
b
a
Which one of the following diagrams illustrates the relation betweena andb andr , the
resultant vector?
A)
a
b
r
C)
a
b
r
B)
a
b
r
D)
a
r
b
11. Operations on Vectors
• Vectors can moved around.
• They must be added tip to tail.
• The resultant of u+v = v+u
• If u = (a,b), and v = (c,d), then
• u + v = (a+c, b+d)
• u- v= (a-c, b-d)
• The multiplication by k of a vector, v
• kv = (kc, kd), ║kv║= ║k║v║
12. Scalar (Dot) Product
• The scalar product of two vectors,
• uv where u = (a,b), v = (c,d)
• = ac + bd = ║u║v║cosΘ,
• Where Θ is the angle between the 2 vector
tails.
• When Θ = 90˚, cos Θ = 0, so the scalar
product is 0.
• Therefore the scalar product of two
perpendicular vectors is 0.
13. Multiplication of a Vector
• The product of a vector and a scalar is
always a vector.
• Associative: a(bv) = (ab)v
• The existence of the scalar, k=1,
is used as an identity element, so that
1v= v
• Distributivity: a(u+v) = au+av
• Distributivity: (a+b)v = av+bv
14. Exam Question
Given the following information:
a andb are nonzero vectors in the plane
ba
k is a scalar not equal to zero
k 1
Which of the following statements is true?
A) bkak)bak(
B) bkak)bak(
C) If
0ba
thena andb are collinear
D) Ifa = kb thena andb are noncollinear
15. Basis Vectors
• Any vector can be broken down into a sum of 2
other vectors, which in turn can be broken down
into a product of a scalar and a vector.
• A linear combination defines a vector by using
other vectors defined previously.
• The real numbers used are the coefficients of
the combination.
• Vector (3,4) = (3,0) + (0,4)
• =3(1,0) + 4(0,1)
• = 3 i + 4 j
• Where i = (1,0) and j = (0,1), coefficients 3,4
16. Exam Question
The Egyptians used an ingenious pulley system to move the blocks of stone used in the
construction of pyramids. To minimize the work needed to displace the blocks, they
applied a force oriented at 26. (Work (Nm) is the scalar product of the force vector and
the displacement vector.)
26
200 m
1500 N
Rounded to the nearest Nm, what work is needed to displace a block of stone horizontally
for a distance of 200 m, if the force applied to it is 1500 N oriented at 26o
?
A) 131 511 Nm C) 228 768 Nm
B) 194 076 Nm D) 269 638 Nm
18. Exam Question
Given )4-,1(vand),2,3(u
What are the components of the resultant of the following vector operation?
v2u
A) (1, 10) C) (2, 6)
B) (1, -6) D) (5, -6)
19. Exam Question
Given the following prism having a rectangular
base.
A B
D C
E F
H
G
Which vector is equivalent to the resultant of the expressionAD +HE +AE ?
A) DH C) FB
B) BE D) BC
21. Summary
• A vector is a quantity that has a size and a
direction and is represented by an arrow
representing its characteristics.
• AB corresponds with –BA
• The size of AB, called its magnitude, or
norm, is written ║AB║
• ║AB║= √((x2-x1)2 + (y2-y1)2)
• If v (c,d),║v║= √((c)2 + (d)2)
22. Summary
• Zero vector is a vector with magnitude 0.
• It is written 0
• Two vectors can be equal, opposite,
collinear (in-line) or non-collinear, or
orthogonal (perpendicular).
• A resultant is the result of combining two
or more vectors, tip to tail.
• Subtracting a vector equals adding its
opposite.
23. Summary
• Chasles’ Rule, AB + BC = AC
• If u = (a,b), and v = (c,d), then
• u + v = (a+c, b+d)
• u- v= (a-c, b-d)
• The multiplication by k of a vector, v
• kv = (kc, kd), ║kv║= ║k║v║
• The scalar product of two vectors uv = ac
+ bd = ║u║v║cosΘ,
• Where Θ is the angle between the 2 vector tails.
24. Summary
• The basis of a the Cartesian plane is x
and y. This can also be expressed as 2
vectors i (1,0) and j (0,1). These
vectors are orthogonal or perpendicular to
each other.
• Other vectors may for the basis for other
vectors.